$x^2+y^3=z^6$ Generalized Fermat Equation

hbghlyj

$y^3=(z^3-x)(z^3+x)$
$\gcd(z,x)=1\implies\gcd(z^3-x,z^3+x)\mid2$
1° $z^3-x=a^3,z^3+x=b^3\implies a^3+b^3=2z^3$
2° $z^3-x=2a^3,z^3+x=4b^3\implies z^3+(-a)^3=2b^3$

hbghlyj

$a^3+b^3=2z^3$
$\gcd(a,b)=1\implies 2\nmid a,b$.
$u:=\frac{a+b}2,v:=\frac{a-b}2\implies u (u^2+3v^2)= z^3$
$\implies u = r^3,u^2+3v^2= s^3$
$(u+\sqrt{-3}v)(u-\sqrt{-3}v)= s^3$
NumberFieldClassNumber($\sqrt{-3}$)=1
$\implies u+\sqrt{-3}v=a_0^3,u-\sqrt{-3}v=b_0^3,a_0^3+b_0^3=2r^3,|r|<|z|$

接下来不知应该怎么做，已在MSE提问：math.stackexchange.com/questions/4896804/ques … olution-for-x2-y3-z6

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hbghlyj

H. Cohen, Number theory, Volume II: Analytic and modern tools, Graduate Texts in Mathematics 240, Springer, New York, 2007.